{
  "abstract": "We resolve three conjectures proposed by Tokushige in 2013 about cross $t$-intersecting families of subsets and integer sequences. For $0<p<1$ and $\\mathcal F\\subseteq2^{[n]}$, define the $p$-biased measure by $μ_p(\\mathcal F)=\\sum_{F\\in\\mathcal F}p^{|F|}(1-p)^{n-|F|}$. Two families $\\mathcal F_1,\\mathcal F_2\\subseteq2^{[n]}$ are cross $t$-intersecting if $|F_1\\cap F_2|\\geq t$ for every $F_1\\in\\mathcal F_1$ and $F_2\\in\\mathcal F_2$. We prove that, for every $n\\geq t\\geq 2$ and $0<p_1,p_2\\leq1/(t+1)$, such a pair satisfies $μ_{p_1}(\\mathcal F_1)μ_{p_2}(\\mathcal F_2)\\leq(p_1p_2)^t$. When $p_1,p_2<1/(t+1)$, equality holds if and only if both families are the same $t$-star $\\mathcal S_T=\\{F\\subseteq[n]:T\\subseteq F\\}$ for some $T\\in\\binom{[n]}{t}$. Together with the previously known case $t=1$, this completely resolves Tokushige's measure conjecture. We also prove that every pair whose measure product is close to the maximum must be close to a common $t$-star. More precisely, if $p_1,p_2<1/(t+1)$ and $μ_{p_1}(\\mathcal F_1)μ_{p_2}(\\mathcal F_2)>(1-\\varepsilon)^2(p_1p_2)^t$, then there exists $T\\in\\binom{[n]}{t}$ such that $μ_{p_i}(\\mathcal F_i\\mathbin{\\triangle}\\mathcal S_T)<C\\varepsilon$ for $i=1,2$, where $C$ depends only on $t,p_1,p_2$. This improves Tokushige's conjectured $C\\sqrt{\\varepsilon}$ estimate to $C\\varepsilon$. For integer sequences, we prove that if every sequence in $\\mathcal H_1\\subseteq[m]^n$ agrees with every sequence in $\\mathcal H_2\\subseteq[m]^n$ in at least $t$ coordinates, then $|\\mathcal H_1||\\mathcal H_2|\\leq m^{2(n-t)}$ for all $n\\geq t\\geq1$ and $m\\geq t+1$. We further obtain a more general result in which a separate agreement requirement is imposed for each possible value. This extends a theorem of Frankl and Kupavskii and recovers their earlier cross intersection--union product theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29512",
  "author_total": 2,
  "authors": [
    {
      "name": "Yongjiang Wu"
    },
    {
      "name": "Lihua Feng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4zgtk432kpgtf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29512",
  "fulltext_url": null,
  "id": "4zgtk432kpgtf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29512",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7204949219",
  "pdf_url": "https://arxiv.org/pdf/2608.29512",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4zgtk432kpgtf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29512",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29512"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A complete solution to the Tokushige measure conjecture and its stability",
  "title_text": "A complete solution to the Tokushige measure conjecture and its stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/4zgtk432kpgtf",
  "venue": "arXiv",
  "versions": []
}